Problem 11 · 2009 Math Kangaroo
Stretch
Algebra & Patterns
work-backwardsubstitution
Leonhard made up a sequence in which, starting with the third term, each term is the sum of the previous two terms. The fourth term is 6 and the sixth term is 15. What is the seventh term of the sequence?
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Answer: E — 24
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Hint 1 of 2
Each term is the sum of the two before it; write the 4th and 6th in terms of the first two.
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Hint 2 of 2
Two equations in the first two terms pin everything down.
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Approach: set up the recurrence from the two known terms
- Let the first two terms be a and b. Then t₄ = a + 2b = 6 and t₆ = 3a + 5b = 15.
- Solving gives b = 3 and a = 0, so the sequence is 0, 3, 3, 6, 9, 15, …
- The seventh term is 9 + 15 = 24.
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